We examine ice crystallization from liquid water and from water vapor, focusing on the underlying physical processes that determine growth rates and structure formation. Ice crystal growth is largely controlled by a combination of molecular attachment kinetics on faceted surfaces and large-scale diffusion processes, yielding a remarkably rich phenomenology of solidification behaviors under different conditions. Layer nucleation plays an especially important role, with nucleation rates determined primarily by step energies on faceted ice/water and ice/vapor interfaces. The measured step energies depend strongly on temperature and other factors, and it appears promising that molecular dynamics simulations could soon be used in conjunction with experiments to better understand the energetics of these terrace steps. On larger scales, computational techniques have recently demonstrated the ability to accurately model the diffusion-limited growth of complex structures that are both faceted and branched. Together with proper boundary conditions determined by surface attachment kinetics, this opens a path to fully reproducing the variety of complex structures that commonly arise during ice crystal growth.